A minimax inequality for a class of functionals and applications to the existence of solutions for two-point boundary-value problems

In this paper, we establish an equivalent statement to minimax inequality for a special class of functionals. As an application, we prove the existence of three solutions to the Dirichlet problem $$displaylines{ -u''(x)+m(x)u(x) =lambda f(x,u(x)),quad xin (a,b),cr u(a)=u(b)=0, }$$ where $lambda>0$, $f:[a,b]imes mathbb{R}o mathbb{R}$ is a continuous function which changes sign on $[a,b]imes mathbb{R}$ and $m(x)in C([a,b])$ is a positive function.